Optimization problems in financial economics (06w5028)

Objectives

We propose a workshop that would focus on several aspects of optimization problems appearing in finance and economics, and the questions that these raise in terms of mathematical theories. We focus our attention on non linear and/or high-dimensional optimization problems that are commonly encountered in portfolio allocation, asset pricing and contract theory. We aim to organize a workshop where the recent theoretical and numerical methods on optimizations methods are applied in meaningful financial economics problems.

While many theoretical and numerical advances have been recently realized in the field of optimization techniques such as the Forward Backward Stochastic Differential Equations, it seems that the potential applicability in economics and finance remains to be done. It is our beliefs that the workshop would be timely because the above mentioned advances open the door for new economic and financial applications and a workshop would give the opportunity to go in these directions. It would greatly help the process of advancing the above topics if a group of leading researchers in this and related fields would get together for a period of time and try to advance with synergy.

We propose to invite a group that would include experts on numerical methods, PDE's, Monte Carlo simulation, quantitative finance, Malliavin Calculus, Forward Backward Stochastic Differential Equations. Such a group would consist mostly of mathematicians, but also representative researchers from finance and economic departments, and a few from finance industry. We will balance the number of participants between senior scientists, and promising young researchers as well as doctoral and postdoctoral students. The format of the workshop would contain some talks in the morning and in the beginning of the afternoon. We would free the rest of the afternoon to let the participants work together in small groups.

Optimization tools we aim to discuss include:

1. FORWARD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS.

Optimization problems in continuous-time financial models are typically equivalent to a system of Forward-Backward Stochastic Differential Equations (FBSDEs), for which the existence theory has not been fully completed. It is possible that a good way to approach the optimization problems numerically is to try to solve the corresponding FBSDE. On a theoretical side, we mention here the papers by El Karoui, Peng and Quenez, (2001) and Schroder and Skiadas (2003), and on numerical side the papers by Bouchard and Touzi (2004), and Zhang (2004). Moreover, Backward SDE's also allow convenient modeling of asymmetric information, by considering models under different filtrations, as in Lazrak (2004). They are also useful in solving classical Principal-Agent problems in economics, in continuous-time models, as in Cvitanic, Wan and Zhang (2004). The aim is to further develop the FBSDE approach to optimization problems in finance, both numerically, and regarding existence and uniqueness of their solutions

2. NUMERICAL METHODS

One of the hardest practical problems of quantitative financial methods is to solve high-dimensional optimization problems. The most famous example of these is pricing high-dimensional American options. In principle, at least in diffusion models, these problems can be solved by solving nonlinear partial differential equations or free boundary problems. However, with many variables (interest rates, volatilities, various stocks and other financial variables), these PDE's are high-dimensional and standard numerical methods do not work.

For linear problems, which correspond to finding expected values rather than a supremum of expected values, high-dimensional problems can be solved, simply by using Monte Carlo simulation. In the last several years some progress has been achieved in trying to apply simulation methods to optimization/nonlinear problems. These are usually based either on nonparametric regression kernel techniques, or on infinite dimensional Malliavin Calculus approach. Representative early papers are Longstaff and Schwartz (2001) and Fournie et al (1999) (the latter dealing with linear problems only). We hope to make some significant progress in this field, with the ultimate aim of developing efficient numerical methods for solving high-dimensional optimization/nonlinear PDE problems with the help of Monte Carlo methods.

The applications we aim to discuss include

1. CONTRACT THEORY

Many of the above mentioned problems become both more theoretically interesting and more practical when considered in a context of two or more market participants. For instance, in the Principal-Agent problems, the principal hires the agent to perform certain tasks (such as managing an investment fund or running a company). The principal must then design a compensation contract which gives to the agent the incentives to realize the maximal effort for the assigned task. The problem becomes even more interesting (and more challenging) in a dynamic context and in presence of asymmetric information in the sense that the agent may have more information than the principal about the underlying activity risk to which the principal is exposed. Finding solution to problems like these leads to better understanding of the functioning of the financial markets, and of optimal ways to reward fund managers or executives.

2. PORTFOLIO ALLOCATION

Perhaps the most classical optimization problem in finance is the problem of optimal portfolio allocation. Many problems of this type have been solved in complete markets explicitly, and in Markovian models of incomplete markets numerically. Still, in high-dimensions numerical methods are not yet satisfactory, and for practical applications it is often useful to have analytic solutions, especially for problems related to risk management and hedging in incomplete markets. Thus, there is a lot of room for improvement. In particular, the recent trends in financial industry call for finding ways for portfolio allocation with imposed limits on the underlying risk measures. Similarly, introducing some information asymmetry and interaction between investors in the portfolio allocation models should make the modeling more realistic and readily available for empirical tests.

References:

- B. Bouchard and N. Touzi, (2004) �Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations�, Stochastic Processes and their Applications, 111, 175-206.

- J. Cvitanic, X. Wan and J. Zhang (2004) " Continuous-time Principal-Agent problems: necessary and sufficient conditions for optimality." Preprint.

- N. El Karoui , S. Peng , M. C. Quenez (2001) "A dynamic maximum principle for the optimization of recursive utilities under constraints". The Annals of Applied Probability 11, 664-693.

- E. Fournie, J.-M. Lasry, J. Lebuchoux, P.L. Lions and N. Touzi (1999) "Applications of Malliavin calculus to Monte Carlo methods in finance". Finance and Stochastics 3, 391-412.

-A. Lazrak (2004) "Generalized Stochastic Differential Utility and Preference for Information", Annals of Applied Probability, to appear.

- F.A. Longstaff and E.S. Schwartz. (2001) "Valuing American options by simulation: a simple least-squares approach". Review of Financial Studies 14, 113-148.

- C. Skiadas and M. Schroder (2003) "Optimal Lifetime Consumption-Portfolio Strategies under Trading Constraints and Generalized Recursive Preferences'' (with M. Schroder), Stochastic Processes and their Applications, 108, 155-202

- J. Zhang (2004), A numerical scheme for backward stochastic differential equations, Annals of Applied Probability, to appear.